arXiv subjects
Publications and source records attributed to V. I. Man'ko.
It is shown that charged-particle beam transport in the paraxial approximation can be effectively described with a quantum-like picture in semiclassical approximation. In particular, the classical Liouville equation can be suitably replaced by a von Neuman-like equation. Relevant remarks concerning the standard classical description of the beam transport are given.
In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, new equations for the Green's function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green's function of the Schrödinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the new definition recovering well known results. As an application, the forced parametric oscillator is considered . Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.
Following the paper by M. Combescure [Ann. Phys. (NY) 204, 113 (1990)], we apply the quantum singular time dependent oscillator model to describe the relative one dimensional motion of two ions in a trap. We argue that the model can be justified for low energy excited states with the quantum numbers $n\ll n_{max}\sim 100$, provided that the dimensionless constant characterizing the strength of the repulsive potential is large enough, $g_*\sim 10^5$. Time dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator, and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center of mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential, since the large value of $g_*$ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.
The characterization of non-stationary signals requires joint time and frequency information. However, time (t) and frequency (omega) being non-commuting variables there cannot be a joint probability density in the (t,omega) plane and the time-frequency distributions, that have been proposed, have difficult interpretation problems arising from negative or complex values and spurious components. As an alternative we propose to obtain time-frequency information by looking at the marginal distributions along rotated directions in the (t,omega) plane. The rigorous probability interpretation of the marginal distributions avoids all interpretation ambiguities. Applications to signal analysis and signal detection are discussed as well as an extension of the method to other pairs of non-commuting variables.
q-oscillators are associated to the simplest non-commutative example of Hopf algebra and may be considered to be the basic building blocks for the symmetry algebras of completely integrable theories. They may also be interpreted as a special type of spectral nonlinearity, which may be generalized to a wider class of f-oscillator algebras. In the framework of this nonlinear interpretation, we discuss the structure of the stochastic process associated to q-deformation, the role of the q-oscillator as a spectrum-generating algebra for fast growing point spectrum, the deformation of fermion operators in solid-state models and the charge-dependent mass of excitations in f-deformed relativistic quantum fields.
The q-deformation of harmonic oscillators is shown to lead to q-nonlinear vibrations. The examples of q-nonlinearized wave equation and Schrödinger equation are considered. The procedure is generalized to broader class of nonlinearities related to other types of deformations. The nonlinear noncanonical transforms used in the deformation procedure are shown to preserve in some cases the linear dynamical equations, for instance, for the harmonic oscillators. The nonlinear coherent states and some physical aspects of the deformations are reviewed.
For Dirac equation, operator-invariants containing explicit time-dependence in parallel to known time-dependent invariants of nonrelativistic Schrödinger equation are introduced and discussed. As an example, a free Dirac particle is considered and new invariants are constructed for it. The integral of motion, which is initial Newton-Wigner position operator, is obtained explicitly for a free Dirac particle. For such particle with kick modeled by delta-function of time, the time-depending integral, which has physical meaning of initial momentum, is found.
The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.
It is shown that because of the radiation pressure a Schrödinger cat state can be generated in a resonator with oscillating wall. The optomechanical control of quantum macroscopic coherence and its detection is taken into account introducing new cat states. The effects due to the environmental couplings with this nonlinear system are considered developing an operator perturbation procedure to solve the master equation for the field mode density operator.
It is shown, that for quantum systems the vectorfield associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrödinger and Heisenberg pictures. We illustrate these ambiguities in terms of simple examples.
The problem of wave packet tunneling from a parabolic potential well through a barrier represented by a power potential is considered in the case when the barrier height is much greater than the oscillator ground state energy, and the difference between the average energy of the packet and the nearest oscillator eigenvalue is sufficiently small. The universal Poisson distribution of the partial tunneling rates from the oscillator energy levels is discovered. The explicit expressions for the tunneling rates of different types of packets (coherent, squeezed, even/odd, thermal, etc.) are given in terms of the exponential and modified Bessel functions. The tunneling rates turn out very sensitive to the energy distributions in the packets, and they may exceed significantly the tunneling rate from the energy state with the same average number of quanta.
The Wigner function for one and two-mode quantum systems is explicitely expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, also the density operator of those systems is written in terms of the marginal distribution of these quadratures. Some examples to apply this formalism, and a reduction to the usual optical homodyne tomography are considered.
By using a generalization of the optical tomography technique we describe the dynamics of a quantum system in terms of equations for a purely classical probability distribution which contains complete information about the system.
Within the Thermal Wave Model framework a comparison among Wigner function, Husimi function, and the phase-space distribution given by a particle tracking code is made for a particle beam travelling through a linear lens with small aberrations. The results show that the quantum-like approach seems to be very promising.
The theory of quantum propagator and time--dependent integrals of motion in quantum optics is reviewed as well as the properties of Wigner function, Q--function, and coherent state representation. Propagators and wave functions of a free particle, harmonic oscillator, and the oscillator with varying frequency are studied using time--dependent linear in position and momentum integrals of motion. Such nonclassical states of light (of quantum systems) as squeezed states, correlated states, even and odd coherent states (Schrödinger cat states) are considered. Photon distribution functions of Schrödinger cat male and female states are given, and the photon distribution function of squeezed vacuum is derived using the theory of the oscillator with varying parameters. Properties of multivariable Hermite polynomials used for the description of the multimode squeezed and correlated light and polymode Schrödinger cats are studied.
We discuss the implications of an experiment in which the frequencies of two laser beams are compared for different intensities in order to search for a dependence of the frequency of light on its intensity. Since no such dependence was found it is possible to place bounds on a description of the electromagnetic field in terms of q-oscillators. We conclude that the value of the nonlinearity parameter is smaller than $10^{-17}~$.
In this paper, by using an analogy among {\it quantum mechanics}, {\it electromagnetic beam optics in optical fibers}, and {\it charge particle beam dynamics}, we introduce the concept of {\it coherent states} for charged particle beams in the framework of the {\it Thermal Wave Model} (TWM). We give a physical meaning of the Gaussian-like coherent structures of charged particle distribution that are both naturally and artificially produced in an accelerating machine in terms of the concept of coherent states widely used in quantum mechanics and in quantum optics. According to TWM, this can be done by using a Schrödinger-like equation for a complex function, the so-called {\it beam wave function} (BWF), whose squared modulus is proportional to the transverse beam density profile, where Planck's constant and the time are replaced by the transverse beam emittance and by the propagation coordinate, respectively. The evolution of the particle beam, whose initial BWF is assumed to be the simplest coherent state (ground-like state) associated with the beam, in an infinite 1-D quadrupole-like device with small sextupole and octupole aberrations, is analytically and numerically investigated.
The primary aim of the present paper is to attract the attention of particle physicists to new developments in studying squeezed and correlated states of the electromagnetic field as well as of those working on the latest topic to new findings about multiplicity distributions in quantum chromodynamics. The new types of nonclassical states used in quantum optics as squeezed states, correlated states, even and odd coherent states (Schrödinger cat states) for one--mode and multimode interaction are reviewed. Their distribution functions are analyzed according to the method used first for multiplicity distributions in high energy particle interactions. The phenomenon of oscillations of particle distribution functions of the squeezed fields is described and confronted to the phenomenon of oscillations of cumulant moments of some distributions for squeezed and correlated field states. Possible extension of the method to fields different from the electromagnetic field (gluons, pions, etc.) is speculated.